Hands-on Exercise 8: Hedonic Pricing model with Geographically Weighted Regression
Import Data
This exercise focuses on building a Hedonic pricing model on condo prices in 2015 with Geographically weighted model.
Geospatial Data Wrangling
Through observation, the imported shapefile is not in the expected coordinate system - SVY21, need to apply st_transform() on the data to convert to the correct coordinate system.
Show the code
Reading layer `MP14_SUBZONE_WEB_PL' from data source
`/Users/tangtang/Desktop/IS415 Geospatial Analytics and Applications/practice/is415gaa/data/geospatial'
using driver `ESRI Shapefile'
Simple feature collection with 323 features and 15 fields
Geometry type: MULTIPOLYGON
Dimension: XY
Bounding box: xmin: 2667.538 ymin: 15748.72 xmax: 56396.44 ymax: 50256.33
Projected CRS: SVY21Coordinate Reference System:
User input: EPSG:3414
wkt:
PROJCRS["SVY21 / Singapore TM",
BASEGEOGCRS["SVY21",
DATUM["SVY21",
ELLIPSOID["WGS 84",6378137,298.257223563,
LENGTHUNIT["metre",1]]],
PRIMEM["Greenwich",0,
ANGLEUNIT["degree",0.0174532925199433]],
ID["EPSG",4757]],
CONVERSION["Singapore Transverse Mercator",
METHOD["Transverse Mercator",
ID["EPSG",9807]],
PARAMETER["Latitude of natural origin",1.36666666666667,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8801]],
PARAMETER["Longitude of natural origin",103.833333333333,
ANGLEUNIT["degree",0.0174532925199433],
ID["EPSG",8802]],
PARAMETER["Scale factor at natural origin",1,
SCALEUNIT["unity",1],
ID["EPSG",8805]],
PARAMETER["False easting",28001.642,
LENGTHUNIT["metre",1],
ID["EPSG",8806]],
PARAMETER["False northing",38744.572,
LENGTHUNIT["metre",1],
ID["EPSG",8807]]],
CS[Cartesian,2],
AXIS["northing (N)",north,
ORDER[1],
LENGTHUNIT["metre",1]],
AXIS["easting (E)",east,
ORDER[2],
LENGTHUNIT["metre",1]],
USAGE[
SCOPE["Cadastre, engineering survey, topographic mapping."],
AREA["Singapore - onshore and offshore."],
BBOX[1.13,103.59,1.47,104.07]],
ID["EPSG",3414]]We can see the size of the boundary in terms of pixels using st_bbox(). It will return the size of the plot in Cartesean coordinates (which truly reflects that of Singapore).
After we are done with pre-processing geospatial data, we need to import aspatial data used in this exercise, which is condo prices in 2015.
Rows: 1436 Columns: 23
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (23): LATITUDE, LONGITUDE, POSTCODE, SELLING_PRICE, AREA_SQM, AGE, PROX_...
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.Rows: 1,436
Columns: 23
$ LATITUDE <dbl> 1.287145, 1.328698, 1.313727, 1.308563, 1.321437,…
$ LONGITUDE <dbl> 103.7802, 103.8123, 103.7971, 103.8247, 103.9505,…
$ POSTCODE <dbl> 118635, 288420, 267833, 258380, 467169, 466472, 3…
$ SELLING_PRICE <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1320…
$ AREA_SQM <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 168,…
$ AGE <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22, 6,…
$ PROX_CBD <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783402…
$ PROX_CHILDCARE <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543, 0…
$ PROX_ELDERLYCARE <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.121…
$ PROX_URA_GROWTH_AREA <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.410632,…
$ PROX_HAWKER_MARKET <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969, 0…
$ PROX_KINDERGARTEN <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076, 0…
$ PROX_MRT <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.528…
$ PROX_PARK <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.116…
$ PROX_PRIMARY_SCH <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.709…
$ PROX_TOP_PRIMARY_SCH <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.709…
$ PROX_SHOPPING_MALL <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.307…
$ PROX_SUPERMARKET <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.581…
$ PROX_BUS_STOP <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340, 0…
$ NO_Of_UNITS <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34, 3…
$ FAMILY_FRIENDLY <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0…
$ FREEHOLD <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…[1] 103.7802 103.8123 103.7971 103.8247 103.9505 103.9386[1] 1.287145 1.328698 1.313727 1.308563 1.321437 1.314198Summary statistics of condo resales prices in 2015
LATITUDE LONGITUDE POSTCODE SELLING_PRICE
Min. :1.240 Min. :103.7 Min. : 18965 Min. : 540000
1st Qu.:1.309 1st Qu.:103.8 1st Qu.:259849 1st Qu.: 1100000
Median :1.328 Median :103.8 Median :469298 Median : 1383222
Mean :1.334 Mean :103.8 Mean :440439 Mean : 1751211
3rd Qu.:1.357 3rd Qu.:103.9 3rd Qu.:589486 3rd Qu.: 1950000
Max. :1.454 Max. :104.0 Max. :828833 Max. :18000000
AREA_SQM AGE PROX_CBD PROX_CHILDCARE
Min. : 34.0 Min. : 0.00 Min. : 0.3869 Min. :0.004927
1st Qu.:103.0 1st Qu.: 5.00 1st Qu.: 5.5574 1st Qu.:0.174481
Median :121.0 Median :11.00 Median : 9.3567 Median :0.258135
Mean :136.5 Mean :12.14 Mean : 9.3254 Mean :0.326313
3rd Qu.:156.0 3rd Qu.:18.00 3rd Qu.:12.6661 3rd Qu.:0.368293
Max. :619.0 Max. :37.00 Max. :19.1804 Max. :3.465726
PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_HAWKER_MARKET PROX_KINDERGARTEN
Min. :0.05451 Min. :0.2145 Min. :0.05182 Min. :0.004927
1st Qu.:0.61254 1st Qu.:3.1643 1st Qu.:0.55245 1st Qu.:0.276345
Median :0.94179 Median :4.6186 Median :0.90842 Median :0.413385
Mean :1.05351 Mean :4.5981 Mean :1.27987 Mean :0.458903
3rd Qu.:1.35122 3rd Qu.:5.7550 3rd Qu.:1.68578 3rd Qu.:0.578474
Max. :3.94916 Max. :9.1554 Max. :5.37435 Max. :2.229045
PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_TOP_PRIMARY_SCH
Min. :0.05278 Min. :0.02906 Min. :0.07711 Min. :0.07711
1st Qu.:0.34646 1st Qu.:0.26211 1st Qu.:0.44024 1st Qu.:1.34451
Median :0.57430 Median :0.39926 Median :0.63505 Median :1.88213
Mean :0.67316 Mean :0.49802 Mean :0.75471 Mean :2.27347
3rd Qu.:0.84844 3rd Qu.:0.65592 3rd Qu.:0.95104 3rd Qu.:2.90954
Max. :3.48037 Max. :2.16105 Max. :3.92899 Max. :6.74819
PROX_SHOPPING_MALL PROX_SUPERMARKET PROX_BUS_STOP NO_Of_UNITS
Min. :0.0000 Min. :0.0000 Min. :0.001595 Min. : 18.0
1st Qu.:0.5258 1st Qu.:0.3695 1st Qu.:0.098356 1st Qu.: 188.8
Median :0.9357 Median :0.5687 Median :0.151710 Median : 360.0
Mean :1.0455 Mean :0.6141 Mean :0.193974 Mean : 409.2
3rd Qu.:1.3994 3rd Qu.:0.7862 3rd Qu.:0.220466 3rd Qu.: 590.0
Max. :3.4774 Max. :2.2441 Max. :2.476639 Max. :1703.0
FAMILY_FRIENDLY FREEHOLD LEASEHOLD_99YR
Min. :0.0000 Min. :0.0000 Min. :0.0000
1st Qu.:0.0000 1st Qu.:0.0000 1st Qu.:0.0000
Median :0.0000 Median :0.0000 Median :0.0000
Mean :0.4868 Mean :0.4227 Mean :0.4882
3rd Qu.:1.0000 3rd Qu.:1.0000 3rd Qu.:1.0000
Max. :1.0000 Max. :1.0000 Max. :1.0000 The condo_resale is currently a tibble data.frame, convert to sf for easier future processing.
Show the code
Simple feature collection with 6 features and 21 fields
Geometry type: POINT
Dimension: XY
Bounding box: xmin: 22085.12 ymin: 29951.54 xmax: 41042.56 ymax: 34546.2
Projected CRS: SVY21 / Singapore TM
# A tibble: 6 × 22
POSTCODE SELLING_PRICE AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE
<dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
1 118635 3000000 309 30 7.94 0.166 2.52
2 288420 3880000 290 32 6.61 0.280 1.93
3 267833 3325000 248 33 6.90 0.429 0.502
4 258380 4250000 127 7 4.04 0.395 1.99
5 467169 1400000 145 28 11.8 0.119 1.12
6 466472 1320000 139 22 10.3 0.125 0.789
# ℹ 15 more variables: PROX_URA_GROWTH_AREA <dbl>, PROX_HAWKER_MARKET <dbl>,
# PROX_KINDERGARTEN <dbl>, PROX_MRT <dbl>, PROX_PARK <dbl>,
# PROX_PRIMARY_SCH <dbl>, PROX_TOP_PRIMARY_SCH <dbl>,
# PROX_SHOPPING_MALL <dbl>, PROX_SUPERMARKET <dbl>, PROX_BUS_STOP <dbl>,
# NO_Of_UNITS <dbl>, FAMILY_FRIENDLY <dbl>, FREEHOLD <dbl>,
# LEASEHOLD_99YR <dbl>, geometry <POINT [m]>Exploratory Data analysis
Look at distribution of selling price
Show the code
It’s right-skewed, make that normal with log transformation.
Plot again to see if the skewed data is normalised. We can see that it’s relatively less skewed.
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We want to look at the distribution of all the numeric continuous variables
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AREA_SQM <- ggplot(data=condo_resale.sf, aes(x= `AREA_SQM`)) +
geom_histogram(bins=20, color="black", fill="light blue")
AGE <- ggplot(data=condo_resale.sf, aes(x= `AGE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_CBD <- ggplot(data=condo_resale.sf, aes(x= `PROX_CBD`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_CHILDCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_CHILDCARE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_ELDERLYCARE <- ggplot(data=condo_resale.sf, aes(x= `PROX_ELDERLYCARE`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_URA_GROWTH_AREA <- ggplot(data=condo_resale.sf,
aes(x= `PROX_URA_GROWTH_AREA`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_HAWKER_MARKET <- ggplot(data=condo_resale.sf, aes(x= `PROX_HAWKER_MARKET`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_KINDERGARTEN <- ggplot(data=condo_resale.sf, aes(x= `PROX_KINDERGARTEN`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_MRT <- ggplot(data=condo_resale.sf, aes(x= `PROX_MRT`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_PARK <- ggplot(data=condo_resale.sf, aes(x= `PROX_PARK`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_PRIMARY_SCH <- ggplot(data=condo_resale.sf, aes(x= `PROX_PRIMARY_SCH`)) +
geom_histogram(bins=20, color="black", fill="light blue")
PROX_TOP_PRIMARY_SCH <- ggplot(data=condo_resale.sf,
aes(x= `PROX_TOP_PRIMARY_SCH`)) +
geom_histogram(bins=20, color="black", fill="light blue")
ggarrange(AREA_SQM, AGE, PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE,
PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN, PROX_MRT,
PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH,
ncol = 3, nrow = 4)
Look at the geospatial distribution of the condo prices
tmap mode set to interactive viewingShow the code
Warning: The shape mpsz is invalid (after reprojection). See sf::st_is_valid$tm_layout
$tm_layout$set.zoom.limits
[1] 11 14
$tm_layout$style
[1] NA
attr(,"class")
[1] "tm"tmap mode set to plottingHedonic Pricing Modelling with linear regression
Simple Linear regression - take in only 1 predictor
Simple linear regression with selling price (y ie. outcome variable) against area (x ie independent predictor).
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Call:
lm(formula = SELLING_PRICE ~ AREA_SQM, data = condo_resale.sf)
Residuals:
Min 1Q Median 3Q Max
-3695815 -391764 -87517 258900 13503875
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -258121.1 63517.2 -4.064 5.09e-05 ***
AREA_SQM 14719.0 428.1 34.381 < 2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 942700 on 1434 degrees of freedom
Multiple R-squared: 0.4518, Adjusted R-squared: 0.4515
F-statistic: 1182 on 1 and 1434 DF, p-value: < 2.2e-16The estimated equation is SELLING_PRICE = -258121.1 + 14719 * AREA_SQM
R-squared of 0.4518 indicates that model built is able to explain about 45.18% of the resale prices.
Since p-value is much smaller than 0.0001, we will reject the null hypothesis that mean is a good estimator of SELLING_PRICE. This will allow us to infer that simple linear regression model above is a good estimator of SELLING_PRICE.
Let’s see this in a scatter plot
Multivariate Linear Regression - take in > 1 predictors to predict price
Before we use this model, we need to ensure that the assumptions for this model type is fulfilled to ensure that result estimated is truely reflective of what is presented in the data.
We need to take note of
- multicollinearity within predictor variables
This can be investigated through correlation plot.
Show the code
FREEHOLD highly correlated with LEASE_99YEAR, exclude LEASE_99YEAR
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condo.mlr <- lm(formula = SELLING_PRICE ~ AREA_SQM + AGE +
PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET + PROX_KINDERGARTEN +
PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH +
PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET +
PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
data=condo_resale.sf)
summary(condo.mlr)
Call:
lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD + PROX_CHILDCARE +
PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_HAWKER_MARKET +
PROX_KINDERGARTEN + PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH +
PROX_TOP_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_SUPERMARKET +
PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
data = condo_resale.sf)
Residuals:
Min 1Q Median 3Q Max
-3475964 -293923 -23069 241043 12260381
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 481728.40 121441.01 3.967 7.65e-05 ***
AREA_SQM 12708.32 369.59 34.385 < 2e-16 ***
AGE -24440.82 2763.16 -8.845 < 2e-16 ***
PROX_CBD -78669.78 6768.97 -11.622 < 2e-16 ***
PROX_CHILDCARE -351617.91 109467.25 -3.212 0.00135 **
PROX_ELDERLYCARE 171029.42 42110.51 4.061 5.14e-05 ***
PROX_URA_GROWTH_AREA 38474.53 12523.57 3.072 0.00217 **
PROX_HAWKER_MARKET 23746.10 29299.76 0.810 0.41782
PROX_KINDERGARTEN 147468.99 82668.87 1.784 0.07466 .
PROX_MRT -314599.68 57947.44 -5.429 6.66e-08 ***
PROX_PARK 563280.50 66551.68 8.464 < 2e-16 ***
PROX_PRIMARY_SCH 180186.08 65237.95 2.762 0.00582 **
PROX_TOP_PRIMARY_SCH 2280.04 20410.43 0.112 0.91107
PROX_SHOPPING_MALL -206604.06 42840.60 -4.823 1.57e-06 ***
PROX_SUPERMARKET -44991.80 77082.64 -0.584 0.55953
PROX_BUS_STOP 683121.35 138353.28 4.938 8.85e-07 ***
NO_Of_UNITS -231.18 89.03 -2.597 0.00951 **
FAMILY_FRIENDLY 140340.77 47020.55 2.985 0.00289 **
FREEHOLD 359913.01 49220.22 7.312 4.38e-13 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 755800 on 1417 degrees of freedom
Multiple R-squared: 0.6518, Adjusted R-squared: 0.6474
F-statistic: 147.4 on 18 and 1417 DF, p-value: < 2.2e-16The estimated equation is SELLING_PRICE = 481728.40 + 12708.32 * AREA_SQM - 24440.82 * AGE - 78669.78 * PROX_CBD - 351617.91 * PROX_CHILDCARE + 171029.42 * PROX_ELDERLYCARE + 38474.53 * PROX_URA_GROWTH_AREA + 23746.10 * PROX_HAWKER_MARKET + 147468.99 * PROX_KINDERGARTEN - 314599.68 * PROX_MRT + 563280.50 * PROX_PARK + 180186.08 * PROX_PRIMARY_SCH + 2280.04 * PROX_TOP_PRIMARY_SCH - 206604.06 * PROX_SHOPPING_MALL - 44991.80 * PROX_SUPERMARKET + 683121.35 * PROX_BUS_STOP - 231.18 * NO_Of_UNITS + 140340.77 * FAMILY_FRIENDLY + 359913.01 * FREEHOLD
R-squared of 0.6518 indicates that model built is able to explain about 65.18% of the resale prices.
Since p-value is much smaller than 0.0001, we will reject the null hypothesis that mean is a good estimator of SELLING_PRICE. This will allow us to infer that multivariate linear regression model above is also another good estimator of SELLING_PRICE.
Publication Quality Table
From the above, we only want to keep statistically significant predictor variables.
Show the code
Model Summary
-----------------------------------------------------------------------------
R 0.807 RMSE 751998.679
R-Squared 0.651 MSE 571471422208.592
Adj. R-Squared 0.647 Coef. Var 43.168
Pred R-Squared 0.638 AIC 42966.758
MAE 414819.628 SBC 43051.072
-----------------------------------------------------------------------------
RMSE: Root Mean Square Error
MSE: Mean Square Error
MAE: Mean Absolute Error
AIC: Akaike Information Criteria
SBC: Schwarz Bayesian Criteria
ANOVA
--------------------------------------------------------------------------------
Sum of
Squares DF Mean Square F Sig.
--------------------------------------------------------------------------------
Regression 1.512586e+15 14 1.080418e+14 189.059 0.0000
Residual 8.120609e+14 1421 571471422208.592
Total 2.324647e+15 1435
--------------------------------------------------------------------------------
Parameter Estimates
-----------------------------------------------------------------------------------------------------------------
model Beta Std. Error Std. Beta t Sig lower upper
-----------------------------------------------------------------------------------------------------------------
(Intercept) 527633.222 108183.223 4.877 0.000 315417.244 739849.200
AREA_SQM 12777.523 367.479 0.584 34.771 0.000 12056.663 13498.382
AGE -24687.739 2754.845 -0.167 -8.962 0.000 -30091.739 -19283.740
PROX_CBD -77131.323 5763.125 -0.263 -13.384 0.000 -88436.469 -65826.176
PROX_CHILDCARE -318472.751 107959.512 -0.084 -2.950 0.003 -530249.889 -106695.613
PROX_ELDERLYCARE 185575.623 39901.864 0.090 4.651 0.000 107302.737 263848.510
PROX_URA_GROWTH_AREA 39163.254 11754.829 0.060 3.332 0.001 16104.571 62221.936
PROX_MRT -294745.107 56916.367 -0.112 -5.179 0.000 -406394.234 -183095.980
PROX_PARK 570504.807 65507.029 0.150 8.709 0.000 442003.938 699005.677
PROX_PRIMARY_SCH 159856.136 60234.599 0.062 2.654 0.008 41697.849 278014.424
PROX_SHOPPING_MALL -220947.251 36561.832 -0.115 -6.043 0.000 -292668.213 -149226.288
PROX_BUS_STOP 682482.221 134513.243 0.134 5.074 0.000 418616.359 946348.082
NO_Of_UNITS -245.480 87.947 -0.053 -2.791 0.005 -418.000 -72.961
FAMILY_FRIENDLY 146307.576 46893.021 0.057 3.120 0.002 54320.593 238294.560
FREEHOLD 350599.812 48506.485 0.136 7.228 0.000 255447.802 445751.821
-----------------------------------------------------------------------------------------------------------------There’s another way to print this same table
| Characteristic | Beta | 95% CI1 | p-value |
|---|---|---|---|
| (Intercept) | 527,633 | 315,417, 739,849 | <0.001 |
| AREA_SQM | 12,778 | 12,057, 13,498 | <0.001 |
| AGE | -24,688 | -30,092, -19,284 | <0.001 |
| PROX_CBD | -77,131 | -88,436, -65,826 | <0.001 |
| PROX_CHILDCARE | -318,473 | -530,250, -106,696 | 0.003 |
| PROX_ELDERLYCARE | 185,576 | 107,303, 263,849 | <0.001 |
| PROX_URA_GROWTH_AREA | 39,163 | 16,105, 62,222 | <0.001 |
| PROX_MRT | -294,745 | -406,394, -183,096 | <0.001 |
| PROX_PARK | 570,505 | 442,004, 699,006 | <0.001 |
| PROX_PRIMARY_SCH | 159,856 | 41,698, 278,014 | 0.008 |
| PROX_SHOPPING_MALL | -220,947 | -292,668, -149,226 | <0.001 |
| PROX_BUS_STOP | 682,482 | 418,616, 946,348 | <0.001 |
| NO_Of_UNITS | -245 | -418, -73 | 0.005 |
| FAMILY_FRIENDLY | 146,308 | 54,321, 238,295 | 0.002 |
| FREEHOLD | 350,600 | 255,448, 445,752 | <0.001 |
| 1 CI = Confidence Interval | |||
When we want to include model statistics, we use this
Show the code
| Characteristic | Beta | 95% CI1 | p-value |
|---|---|---|---|
| (Intercept) | 527,633 | 315,417, 739,849 | <0.001 |
| AREA_SQM | 12,778 | 12,057, 13,498 | <0.001 |
| AGE | -24,688 | -30,092, -19,284 | <0.001 |
| PROX_CBD | -77,131 | -88,436, -65,826 | <0.001 |
| PROX_CHILDCARE | -318,473 | -530,250, -106,696 | 0.003 |
| PROX_ELDERLYCARE | 185,576 | 107,303, 263,849 | <0.001 |
| PROX_URA_GROWTH_AREA | 39,163 | 16,105, 62,222 | <0.001 |
| PROX_MRT | -294,745 | -406,394, -183,096 | <0.001 |
| PROX_PARK | 570,505 | 442,004, 699,006 | <0.001 |
| PROX_PRIMARY_SCH | 159,856 | 41,698, 278,014 | 0.008 |
| PROX_SHOPPING_MALL | -220,947 | -292,668, -149,226 | <0.001 |
| PROX_BUS_STOP | 682,482 | 418,616, 946,348 | <0.001 |
| NO_Of_UNITS | -245 | -418, -73 | 0.005 |
| FAMILY_FRIENDLY | 146,308 | 54,321, 238,295 | 0.002 |
| FREEHOLD | 350,600 | 255,448, 445,752 | <0.001 |
| R² = 0.651; Adjusted R² = 0.647; AIC = 42,967; Statistic = 189; p-value = <0.001; σ = 755,957 | |||
| 1 CI = Confidence Interval | |||
OLS regression
This is making use of another package to help build linear regression models, olsrr
Multicollinearity can be investigated using ols_vif_tol()
Since the values < 10, we can safely assume no multicollinearity exist.
Variables Tolerance VIF
1 AREA_SQM 0.8728554 1.145665
2 AGE 0.7071275 1.414172
3 PROX_CBD 0.6356147 1.573280
4 PROX_CHILDCARE 0.3066019 3.261559
5 PROX_ELDERLYCARE 0.6598479 1.515501
6 PROX_URA_GROWTH_AREA 0.7510311 1.331503
7 PROX_MRT 0.5236090 1.909822
8 PROX_PARK 0.8279261 1.207837
9 PROX_PRIMARY_SCH 0.4524628 2.210126
10 PROX_SHOPPING_MALL 0.6738795 1.483945
11 PROX_BUS_STOP 0.3514118 2.845664
12 NO_Of_UNITS 0.6901036 1.449058
13 FAMILY_FRIENDLY 0.7244157 1.380423
14 FREEHOLD 0.6931163 1.442759Given that linear regression, as the name suggests, test for linear relationship between the variables and SELLING_PRICE. We also want to know if there is non-linear relationship between variables and SELLING_PRICE.
Most of the data poitns are scattered around the 0 line, hence we can safely conclude that the relationships between the dependent variable and independent variables are linear
One of the assumptions of linear regression is the numeric continuous variables have normal distribution.
Residual of the multiple linear regression model (i.e. condo.mlr1) is resemble normal distribution
This is the statistical way of testing for normality
Values of the four tests are way smaller than the alpha value of 0.05. Hence we will reject the null hypothesis and infer that there is statistical evidence that the residual are not normally distributed.
Warning in ks.test.default(y, "pnorm", mean(y), sd(y)): ties should not be
present for the Kolmogorov-Smirnov test-----------------------------------------------
Test Statistic pvalue
-----------------------------------------------
Shapiro-Wilk 0.6856 0.0000
Kolmogorov-Smirnov 0.1366 0.0000
Cramer-von Mises 121.0768 0.0000
Anderson-Darling 67.9551 0.0000
-----------------------------------------------The hedonic model we try to build are using geographically referenced attributes, hence it is also important for us to visualise the residual of the hedonic pricing model.
Show the code
class : SpatialPointsDataFrame
features : 1436
extent : 14940.85, 43352.45, 24765.67, 48382.81 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=1.36666666666667 +lon_0=103.833333333333 +k=1 +x_0=28001.642 +y_0=38744.572 +ellps=WGS84 +towgs84=0,0,0,0,0,0,0 +units=m +no_defs
variables : 23
names : POSTCODE, SELLING_PRICE, AREA_SQM, AGE, PROX_CBD, PROX_CHILDCARE, PROX_ELDERLYCARE, PROX_URA_GROWTH_AREA, PROX_HAWKER_MARKET, PROX_KINDERGARTEN, PROX_MRT, PROX_PARK, PROX_PRIMARY_SCH, PROX_TOP_PRIMARY_SCH, PROX_SHOPPING_MALL, ...
min values : 18965, 540000, 34, 0, 0.386916393, 0.004927023, 0.054508623, 0.214539508, 0.051817113, 0.004927023, 0.052779424, 0.029064164, 0.077106132, 0.077106132, 0, ...
max values : 828833, 1.8e+07, 619, 37, 19.18042832, 3.46572633, 3.949157205, 9.15540001, 5.374348075, 2.229045366, 3.48037319, 2.16104919, 3.928989144, 6.748192062, 3.477433767, ... Visualising using tmap
Graph show signs of spatial autocorrelation.
tmap mode set to interactive viewingShow the code
Warning: The shape mpsz is invalid (after reprojection). See sf::st_is_validVariable(s) "MLR_RES" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.tmap mode set to plottingGeospatial Autocorrelation
Since we see signs, time to handle.
Neighbour list object:
Number of regions: 1436
Number of nonzero links: 66266
Percentage nonzero weights: 3.213526
Average number of links: 46.14624
10 disjoint connected subgraphs
Link number distribution:
1 3 5 7 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
3 3 9 4 3 15 10 19 17 45 19 5 14 29 19 6 35 45 18 47
25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
16 43 22 26 21 11 9 23 22 13 16 25 21 37 16 18 8 21 4 12
45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
8 36 18 14 14 43 11 12 8 13 12 13 4 5 6 12 11 20 29 33
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
15 20 10 14 15 15 11 16 12 10 8 19 12 14 9 8 4 13 11 6
85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104
4 9 4 4 4 6 2 16 9 4 5 9 3 9 4 2 1 2 1 1
105 106 107 108 109 110 112 116 125
1 5 9 2 1 3 1 1 1
3 least connected regions:
193 194 277 with 1 link
1 most connected region:
285 with 125 linksCharacteristics of weights list object:
Neighbour list object:
Number of regions: 1436
Number of nonzero links: 66266
Percentage nonzero weights: 3.213526
Average number of links: 46.14624
10 disjoint connected subgraphs
Link number distribution:
1 3 5 7 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
3 3 9 4 3 15 10 19 17 45 19 5 14 29 19 6 35 45 18 47
25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
16 43 22 26 21 11 9 23 22 13 16 25 21 37 16 18 8 21 4 12
45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
8 36 18 14 14 43 11 12 8 13 12 13 4 5 6 12 11 20 29 33
65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84
15 20 10 14 15 15 11 16 12 10 8 19 12 14 9 8 4 13 11 6
85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104
4 9 4 4 4 6 2 16 9 4 5 9 3 9 4 2 1 2 1 1
105 106 107 108 109 110 112 116 125
1 5 9 2 1 3 1 1 1
3 least connected regions:
193 194 277 with 1 link
1 most connected region:
285 with 125 links
Weights style: W
Weights constants summary:
n nn S0 S1 S2
W 1436 2062096 1436 94.81916 5798.341
Global Moran I for regression residuals
data:
model: lm(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA + PROX_MRT +
PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP +
NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD, data = condo_resale.sf)
weights: nb_lw
Moran I statistic standard deviate = 24.366, p-value < 2.2e-16
alternative hypothesis: greater
sample estimates:
Observed Moran I Expectation Variance
1.438876e-01 -5.487594e-03 3.758259e-05 The Global Moran’s I test for residual spatial autocorrelation shows that it’s p-value is less than 0.00000000000000022 which is less than the alpha value of 0.05. Hence, we will reject the null hypothesis that the residuals are randomly distributed.
Since the Observed Global Moran I = 0.1424418 which is greater than 0, we can infer than the residuals resemble cluster distribution.
Hedonic Pricing model with olsrr
Fixed bandwidth GWR
Show the code
bw.fixed <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA +
PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH +
PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS +
FAMILY_FRIENDLY + FREEHOLD,
data=condo_resale.sp,
approach="CV",
kernel="gaussian",
adaptive=FALSE,
longlat=FALSE)Fixed bandwidth: 17660.96 CV score: 8.259118e+14
Fixed bandwidth: 10917.26 CV score: 7.970454e+14
Fixed bandwidth: 6749.419 CV score: 7.273273e+14
Fixed bandwidth: 4173.553 CV score: 6.300006e+14
Fixed bandwidth: 2581.58 CV score: 5.404958e+14
Fixed bandwidth: 1597.687 CV score: 4.857515e+14
Fixed bandwidth: 989.6077 CV score: 4.722431e+14
Fixed bandwidth: 613.7939 CV score: 1.379526e+16
Fixed bandwidth: 1221.873 CV score: 4.778717e+14
Fixed bandwidth: 846.0596 CV score: 4.791629e+14
Fixed bandwidth: 1078.325 CV score: 4.751406e+14
Fixed bandwidth: 934.7772 CV score: 4.72518e+14
Fixed bandwidth: 1023.495 CV score: 4.730305e+14
Fixed bandwidth: 968.6643 CV score: 4.721317e+14
Fixed bandwidth: 955.7206 CV score: 4.722072e+14
Fixed bandwidth: 976.6639 CV score: 4.721387e+14
Fixed bandwidth: 963.7202 CV score: 4.721484e+14
Fixed bandwidth: 971.7199 CV score: 4.721293e+14
Fixed bandwidth: 973.6083 CV score: 4.721309e+14
Fixed bandwidth: 970.5527 CV score: 4.721295e+14
Fixed bandwidth: 972.4412 CV score: 4.721296e+14
Fixed bandwidth: 971.2741 CV score: 4.721292e+14
Fixed bandwidth: 970.9985 CV score: 4.721293e+14
Fixed bandwidth: 971.4443 CV score: 4.721292e+14
Fixed bandwidth: 971.5496 CV score: 4.721293e+14
Fixed bandwidth: 971.3793 CV score: 4.721292e+14
Fixed bandwidth: 971.3391 CV score: 4.721292e+14
Fixed bandwidth: 971.3143 CV score: 4.721292e+14
Fixed bandwidth: 971.3545 CV score: 4.721292e+14
Fixed bandwidth: 971.3296 CV score: 4.721292e+14
Fixed bandwidth: 971.345 CV score: 4.721292e+14
Fixed bandwidth: 971.3355 CV score: 4.721292e+14
Fixed bandwidth: 971.3413 CV score: 4.721292e+14
Fixed bandwidth: 971.3377 CV score: 4.721292e+14
Fixed bandwidth: 971.34 CV score: 4.721292e+14
Fixed bandwidth: 971.3405 CV score: 4.721292e+14
Fixed bandwidth: 971.3396 CV score: 4.721292e+14
Fixed bandwidth: 971.3402 CV score: 4.721292e+14
Fixed bandwidth: 971.3398 CV score: 4.721292e+14
Fixed bandwidth: 971.34 CV score: 4.721292e+14
Fixed bandwidth: 971.3399 CV score: 4.721292e+14
Fixed bandwidth: 971.34 CV score: 4.721292e+14 Recommended bandwidth: 971.34
Show the code
gwr.fixed <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA +
PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH +
PROX_SHOPPING_MALL + PROX_BUS_STOP + NO_Of_UNITS +
FAMILY_FRIENDLY + FREEHOLD,
data=condo_resale.sp,
bw=bw.fixed,
kernel = 'gaussian',
longlat = FALSE)
gwr.fixed ***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2024-03-17 22:25:30.500389
Call:
gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA +
PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL +
PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
data = condo_resale.sp, bw = bw.fixed, kernel = "gaussian",
longlat = FALSE)
Dependent (y) variable: SELLING_PRICE
Independent variables: AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
Number of data points: 1436
***********************************************************************
* Results of Global Regression *
***********************************************************************
Call:
lm(formula = formula, data = data)
Residuals:
Min 1Q Median 3Q Max
-3470778 -298119 -23481 248917 12234210
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 527633.22 108183.22 4.877 1.20e-06 ***
AREA_SQM 12777.52 367.48 34.771 < 2e-16 ***
AGE -24687.74 2754.84 -8.962 < 2e-16 ***
PROX_CBD -77131.32 5763.12 -13.384 < 2e-16 ***
PROX_CHILDCARE -318472.75 107959.51 -2.950 0.003231 **
PROX_ELDERLYCARE 185575.62 39901.86 4.651 3.61e-06 ***
PROX_URA_GROWTH_AREA 39163.25 11754.83 3.332 0.000885 ***
PROX_MRT -294745.11 56916.37 -5.179 2.56e-07 ***
PROX_PARK 570504.81 65507.03 8.709 < 2e-16 ***
PROX_PRIMARY_SCH 159856.14 60234.60 2.654 0.008046 **
PROX_SHOPPING_MALL -220947.25 36561.83 -6.043 1.93e-09 ***
PROX_BUS_STOP 682482.22 134513.24 5.074 4.42e-07 ***
NO_Of_UNITS -245.48 87.95 -2.791 0.005321 **
FAMILY_FRIENDLY 146307.58 46893.02 3.120 0.001845 **
FREEHOLD 350599.81 48506.48 7.228 7.98e-13 ***
---Significance stars
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 756000 on 1421 degrees of freedom
Multiple R-squared: 0.6507
Adjusted R-squared: 0.6472
F-statistic: 189.1 on 14 and 1421 DF, p-value: < 2.2e-16
***Extra Diagnostic information
Residual sum of squares: 8.120609e+14
Sigma(hat): 752522.9
AIC: 42966.76
AICc: 42967.14
BIC: 41731.39
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: gaussian
Fixed bandwidth: 971.34
Regression points: the same locations as observations are used.
Distance metric: Euclidean distance metric is used.
****************Summary of GWR coefficient estimates:******************
Min. 1st Qu. Median 3rd Qu.
Intercept -3.5988e+07 -5.1998e+05 7.6780e+05 1.7412e+06
AREA_SQM 1.0003e+03 5.2758e+03 7.4740e+03 1.2301e+04
AGE -1.3475e+05 -2.0813e+04 -8.6260e+03 -3.7784e+03
PROX_CBD -7.7047e+07 -2.3608e+05 -8.3599e+04 3.4646e+04
PROX_CHILDCARE -6.0097e+06 -3.3667e+05 -9.7426e+04 2.9007e+05
PROX_ELDERLYCARE -3.5001e+06 -1.5970e+05 3.1970e+04 1.9577e+05
PROX_URA_GROWTH_AREA -3.0170e+06 -8.2013e+04 7.0749e+04 2.2612e+05
PROX_MRT -3.5282e+06 -6.5836e+05 -1.8833e+05 3.6922e+04
PROX_PARK -1.2062e+06 -2.1732e+05 3.5383e+04 4.1335e+05
PROX_PRIMARY_SCH -2.2695e+07 -1.7066e+05 4.8472e+04 5.1555e+05
PROX_SHOPPING_MALL -7.2585e+06 -1.6684e+05 -1.0517e+04 1.5923e+05
PROX_BUS_STOP -1.4676e+06 -4.5207e+04 3.7601e+05 1.1664e+06
NO_Of_UNITS -1.3170e+03 -2.4822e+02 -3.0846e+01 2.5496e+02
FAMILY_FRIENDLY -2.2749e+06 -1.1140e+05 7.6214e+03 1.6107e+05
FREEHOLD -9.2067e+06 3.8074e+04 1.5169e+05 3.7528e+05
Max.
Intercept 112794435
AREA_SQM 21575
AGE 434203
PROX_CBD 2704604
PROX_CHILDCARE 1654086
PROX_ELDERLYCARE 38867861
PROX_URA_GROWTH_AREA 78515805
PROX_MRT 3124325
PROX_PARK 18122439
PROX_PRIMARY_SCH 4637517
PROX_SHOPPING_MALL 1529953
PROX_BUS_STOP 11342209
NO_Of_UNITS 12907
FAMILY_FRIENDLY 1720745
FREEHOLD 6073642
************************Diagnostic information*************************
Number of data points: 1436
Effective number of parameters (2trace(S) - trace(S'S)): 438.3807
Effective degrees of freedom (n-2trace(S) + trace(S'S)): 997.6193
AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 42263.61
AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41632.36
BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 42515.71
Residual sum of squares: 2.534069e+14
R-square value: 0.8909912
Adjusted R-square value: 0.8430418
***********************************************************************
Program stops at: 2024-03-17 22:25:31.541227 AICc of the gwr is 42263.61 which is significantly smaller than the globel multiple linear regression model of 42967.1
Adaptive bandwidth GWR
Show the code
bw.adaptive <- bw.gwr(formula = SELLING_PRICE ~ AREA_SQM + AGE +
PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK +
PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP +
NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
data=condo_resale.sp,
approach="CV",
kernel="gaussian",
adaptive=TRUE,
longlat=FALSE)Adaptive bandwidth: 895 CV score: 7.952401e+14
Adaptive bandwidth: 561 CV score: 7.667364e+14
Adaptive bandwidth: 354 CV score: 6.953454e+14
Adaptive bandwidth: 226 CV score: 6.15223e+14
Adaptive bandwidth: 147 CV score: 5.674373e+14
Adaptive bandwidth: 98 CV score: 5.426745e+14
Adaptive bandwidth: 68 CV score: 5.168117e+14
Adaptive bandwidth: 49 CV score: 4.859631e+14
Adaptive bandwidth: 37 CV score: 4.646518e+14
Adaptive bandwidth: 30 CV score: 4.422088e+14
Adaptive bandwidth: 25 CV score: 4.430816e+14
Adaptive bandwidth: 32 CV score: 4.505602e+14
Adaptive bandwidth: 27 CV score: 4.462172e+14
Adaptive bandwidth: 30 CV score: 4.422088e+14 Recommended number of datapoints in each bandwidth: 30
Show the code
gwr.adaptive <- gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE +
PROX_CBD + PROX_CHILDCARE + PROX_ELDERLYCARE +
PROX_URA_GROWTH_AREA + PROX_MRT + PROX_PARK +
PROX_PRIMARY_SCH + PROX_SHOPPING_MALL + PROX_BUS_STOP +
NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
data=condo_resale.sp, bw=bw.adaptive,
kernel = 'gaussian',
adaptive=TRUE,
longlat = FALSE)
gwr.adaptive ***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2024-03-17 22:25:39.174313
Call:
gwr.basic(formula = SELLING_PRICE ~ AREA_SQM + AGE + PROX_CBD +
PROX_CHILDCARE + PROX_ELDERLYCARE + PROX_URA_GROWTH_AREA +
PROX_MRT + PROX_PARK + PROX_PRIMARY_SCH + PROX_SHOPPING_MALL +
PROX_BUS_STOP + NO_Of_UNITS + FAMILY_FRIENDLY + FREEHOLD,
data = condo_resale.sp, bw = bw.adaptive, kernel = "gaussian",
adaptive = TRUE, longlat = FALSE)
Dependent (y) variable: SELLING_PRICE
Independent variables: AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
Number of data points: 1436
***********************************************************************
* Results of Global Regression *
***********************************************************************
Call:
lm(formula = formula, data = data)
Residuals:
Min 1Q Median 3Q Max
-3470778 -298119 -23481 248917 12234210
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 527633.22 108183.22 4.877 1.20e-06 ***
AREA_SQM 12777.52 367.48 34.771 < 2e-16 ***
AGE -24687.74 2754.84 -8.962 < 2e-16 ***
PROX_CBD -77131.32 5763.12 -13.384 < 2e-16 ***
PROX_CHILDCARE -318472.75 107959.51 -2.950 0.003231 **
PROX_ELDERLYCARE 185575.62 39901.86 4.651 3.61e-06 ***
PROX_URA_GROWTH_AREA 39163.25 11754.83 3.332 0.000885 ***
PROX_MRT -294745.11 56916.37 -5.179 2.56e-07 ***
PROX_PARK 570504.81 65507.03 8.709 < 2e-16 ***
PROX_PRIMARY_SCH 159856.14 60234.60 2.654 0.008046 **
PROX_SHOPPING_MALL -220947.25 36561.83 -6.043 1.93e-09 ***
PROX_BUS_STOP 682482.22 134513.24 5.074 4.42e-07 ***
NO_Of_UNITS -245.48 87.95 -2.791 0.005321 **
FAMILY_FRIENDLY 146307.58 46893.02 3.120 0.001845 **
FREEHOLD 350599.81 48506.48 7.228 7.98e-13 ***
---Significance stars
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 756000 on 1421 degrees of freedom
Multiple R-squared: 0.6507
Adjusted R-squared: 0.6472
F-statistic: 189.1 on 14 and 1421 DF, p-value: < 2.2e-16
***Extra Diagnostic information
Residual sum of squares: 8.120609e+14
Sigma(hat): 752522.9
AIC: 42966.76
AICc: 42967.14
BIC: 41731.39
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: gaussian
Adaptive bandwidth: 30 (number of nearest neighbours)
Regression points: the same locations as observations are used.
Distance metric: Euclidean distance metric is used.
****************Summary of GWR coefficient estimates:******************
Min. 1st Qu. Median 3rd Qu.
Intercept -1.3487e+08 -2.4669e+05 7.7928e+05 1.6194e+06
AREA_SQM 3.3188e+03 5.6285e+03 7.7825e+03 1.2738e+04
AGE -9.6746e+04 -2.9288e+04 -1.4043e+04 -5.6119e+03
PROX_CBD -2.5330e+06 -1.6256e+05 -7.7242e+04 2.6624e+03
PROX_CHILDCARE -1.2790e+06 -2.0175e+05 8.7158e+03 3.7778e+05
PROX_ELDERLYCARE -1.6212e+06 -9.2050e+04 6.1029e+04 2.8184e+05
PROX_URA_GROWTH_AREA -7.2686e+06 -3.0350e+04 4.5869e+04 2.4613e+05
PROX_MRT -4.3781e+07 -6.7282e+05 -2.2115e+05 -7.4593e+04
PROX_PARK -2.9020e+06 -1.6782e+05 1.1601e+05 4.6572e+05
PROX_PRIMARY_SCH -8.6418e+05 -1.6627e+05 -7.7853e+03 4.3222e+05
PROX_SHOPPING_MALL -1.8272e+06 -1.3175e+05 -1.4049e+04 1.3799e+05
PROX_BUS_STOP -2.0579e+06 -7.1461e+04 4.1104e+05 1.2071e+06
NO_Of_UNITS -2.1993e+03 -2.3685e+02 -3.4699e+01 1.1657e+02
FAMILY_FRIENDLY -5.9879e+05 -5.0927e+04 2.6173e+04 2.2481e+05
FREEHOLD -1.6340e+05 4.0765e+04 1.9023e+05 3.7960e+05
Max.
Intercept 18758355
AREA_SQM 23064
AGE 13303
PROX_CBD 11346650
PROX_CHILDCARE 2892127
PROX_ELDERLYCARE 2465671
PROX_URA_GROWTH_AREA 7384059
PROX_MRT 1186242
PROX_PARK 2588497
PROX_PRIMARY_SCH 3381462
PROX_SHOPPING_MALL 38038564
PROX_BUS_STOP 12081592
NO_Of_UNITS 1010
FAMILY_FRIENDLY 2072414
FREEHOLD 1813995
************************Diagnostic information*************************
Number of data points: 1436
Effective number of parameters (2trace(S) - trace(S'S)): 350.3088
Effective degrees of freedom (n-2trace(S) + trace(S'S)): 1085.691
AICc (GWR book, Fotheringham, et al. 2002, p. 61, eq 2.33): 41982.22
AIC (GWR book, Fotheringham, et al. 2002,GWR p. 96, eq. 4.22): 41546.74
BIC (GWR book, Fotheringham, et al. 2002,GWR p. 61, eq. 2.34): 41914.08
Residual sum of squares: 2.528227e+14
R-square value: 0.8912425
Adjusted R-square value: 0.8561185
***********************************************************************
Program stops at: 2024-03-17 22:25:40.275582 AICc the adaptive distance gwr is 41982.22 which is even smaller than the AICc of the fixed distance gwr of 42263.61.
Visualising GWR output
Show the code
Simple feature collection with 1436 features and 51 fields
Geometry type: POINT
Dimension: XY
Bounding box: xmin: 14940.85 ymin: 24765.67 xmax: 43352.45 ymax: 48382.81
Projected CRS: SVY21 / Singapore TM
First 10 features:
Intercept AREA_SQM AGE PROX_CBD PROX_CHILDCARE PROX_ELDERLYCARE
1 2050011.7 9561.892 -9514.634 -120681.9 319266.92 -393417.79
2 1633128.2 16576.853 -58185.479 -149434.2 441102.18 325188.74
3 3433608.2 13091.861 -26707.386 -259397.8 -120116.82 535855.81
4 234358.9 20730.601 -93308.988 2426853.7 480825.28 314783.72
5 2285804.9 6722.836 -17608.018 -316835.5 90764.78 -137384.61
6 -3568877.4 6039.581 -26535.592 327306.1 -152531.19 -700392.85
7 -2874842.4 16843.575 -59166.727 -983577.2 -177810.50 -122384.02
8 2038086.0 6905.135 -17681.897 -285076.6 70259.40 -96012.78
9 1718478.4 9580.703 -14401.128 105803.4 -657698.02 -123276.00
10 3457054.0 14072.011 -31579.884 -234895.4 79961.45 548581.04
PROX_URA_GROWTH_AREA PROX_MRT PROX_PARK PROX_PRIMARY_SCH
1 -159980.20 -299742.96 -172104.47 242668.03
2 -142290.39 -2510522.23 523379.72 1106830.66
3 -253621.21 -936853.28 209099.85 571462.33
4 -2679297.89 -2039479.50 -759153.26 3127477.21
5 303714.81 -44567.05 -10284.62 30413.56
6 -28051.25 733566.47 1511488.92 320878.23
7 1397676.38 -2745430.34 710114.74 1786570.95
8 269368.71 -14552.99 73533.34 53359.73
9 -361974.72 -476785.32 -132067.59 -40128.92
10 -150024.38 -1503835.53 574155.47 108996.67
PROX_SHOPPING_MALL PROX_BUS_STOP NO_Of_UNITS FAMILY_FRIENDLY FREEHOLD
1 300881.390 1210615.4 104.8290640 -9075.370 303955.6
2 -87693.378 1843587.2 -288.3441183 310074.664 396221.3
3 -126732.712 1411924.9 -9.5532945 5949.746 168821.7
4 -29593.342 7225577.5 -161.3551620 1556178.531 1212515.6
5 -7490.586 677577.0 42.2659674 58986.951 328175.2
6 258583.881 1086012.6 -214.3671271 201992.641 471873.1
7 -384251.210 5094060.5 -0.9212521 359659.512 408871.9
8 -39634.902 735767.1 30.1741069 55602.506 347075.0
9 276718.757 2815772.4 675.1615559 -30453.297 503872.8
10 -454726.822 2123557.0 -21.3044311 -100935.586 213324.6
y yhat residual CV_Score Stud_residual Intercept_SE AREA_SQM_SE
1 3000000 2886532 113468.16 0 0.38207013 516105.5 823.2860
2 3880000 3466801 413198.52 0 1.01433140 488083.5 825.2380
3 3325000 3616527 -291527.20 0 -0.83780678 963711.4 988.2240
4 4250000 5435482 -1185481.63 0 -2.84614670 444185.5 617.4007
5 1400000 1388166 11834.26 0 0.03404453 2119620.6 1376.2778
6 1320000 1516702 -196701.95 0 -0.72065801 28572883.7 2348.0091
7 3410000 3266881 143118.77 0 0.41291992 679546.6 893.5893
8 1420000 1431955 -11955.27 0 -0.03033109 2217773.1 1415.2604
9 2025000 1832799 192200.83 0 0.52018109 814281.8 943.8434
10 2550000 2223364 326635.53 0 1.10559735 2410252.0 1271.4073
AGE_SE PROX_CBD_SE PROX_CHILDCARE_SE PROX_ELDERLYCARE_SE
1 5889.782 37411.22 319111.1 120633.34
2 6226.916 23615.06 299705.3 84546.69
3 6510.236 56103.77 349128.5 129687.07
4 6010.511 469337.41 304965.2 127150.69
5 8180.361 410644.47 698720.6 327371.55
6 14601.909 5272846.47 1141599.8 1653002.19
7 8970.629 346164.20 530101.1 148598.71
8 8661.309 438035.69 742532.8 399221.05
9 11791.208 89148.35 704630.7 329683.30
10 9941.980 173532.77 500976.2 281876.74
PROX_URA_GROWTH_AREA_SE PROX_MRT_SE PROX_PARK_SE PROX_PRIMARY_SCH_SE
1 56207.39 185181.3 205499.6 152400.7
2 76956.50 281133.9 229358.7 165150.7
3 95774.60 275483.7 314124.3 196662.6
4 470762.12 279877.1 227249.4 240878.9
5 474339.56 363830.0 364580.9 249087.7
6 5496627.21 730453.2 1741712.0 683265.5
7 371692.97 375511.9 297400.9 344602.8
8 517977.91 423155.4 440984.4 261251.2
9 153436.22 285325.4 304998.4 278258.5
10 239182.57 571355.7 599131.8 331284.8
PROX_SHOPPING_MALL_SE PROX_BUS_STOP_SE NO_Of_UNITS_SE FAMILY_FRIENDLY_SE
1 109268.8 600668.6 218.1258 131474.7
2 98906.8 410222.1 208.9410 114989.1
3 119913.3 464156.7 210.9828 146607.2
4 177104.1 562810.8 361.7767 108726.6
5 301032.9 740922.4 299.5034 160663.7
6 2931208.6 1418333.3 602.5571 331727.0
7 249969.5 821236.4 532.1978 129241.2
8 351634.0 775038.4 338.6777 171895.1
9 289872.7 850095.5 439.9037 220223.4
10 265529.7 631399.2 259.0169 189125.5
FREEHOLD_SE Intercept_TV AREA_SQM_TV AGE_TV PROX_CBD_TV
1 115954.0 3.9720784 11.614302 -1.615447 -3.22582173
2 130110.0 3.3460017 20.087361 -9.344188 -6.32792021
3 141031.5 3.5629010 13.247868 -4.102368 -4.62353528
4 138239.1 0.5276150 33.577223 -15.524302 5.17080808
5 210641.1 1.0784029 4.884795 -2.152474 -0.77155660
6 374347.3 -0.1249043 2.572214 -1.817269 0.06207388
7 182216.9 -4.2305303 18.849348 -6.595605 -2.84136028
8 216649.4 0.9189786 4.879056 -2.041481 -0.65080678
9 220473.7 2.1104224 10.150733 -1.221345 1.18682383
10 206346.2 1.4343123 11.068059 -3.176418 -1.35360852
PROX_CHILDCARE_TV PROX_ELDERLYCARE_TV PROX_URA_GROWTH_AREA_TV PROX_MRT_TV
1 1.00048819 -3.2612693 -2.846248368 -1.61864578
2 1.47178634 3.8462625 -1.848971738 -8.92998600
3 -0.34404755 4.1319138 -2.648105057 -3.40075727
4 1.57665606 2.4756745 -5.691404992 -7.28705261
5 0.12990138 -0.4196596 0.640289855 -0.12249416
6 -0.13361179 -0.4237096 -0.005103357 1.00426206
7 -0.33542751 -0.8235874 3.760298131 -7.31116712
8 0.09462126 -0.2405003 0.520038994 -0.03439159
9 -0.93339393 -0.3739225 -2.359121712 -1.67102293
10 0.15961128 1.9461735 -0.627237944 -2.63204802
PROX_PARK_TV PROX_PRIMARY_SCH_TV PROX_SHOPPING_MALL_TV PROX_BUS_STOP_TV
1 -0.83749312 1.5923022 2.75358842 2.0154464
2 2.28192684 6.7019454 -0.88662640 4.4941192
3 0.66565951 2.9058009 -1.05686949 3.0419145
4 -3.34061770 12.9836105 -0.16709578 12.8383775
5 -0.02820944 0.1220998 -0.02488294 0.9145046
6 0.86781794 0.4696245 0.08821750 0.7656963
7 2.38773567 5.1844351 -1.53719231 6.2029165
8 0.16674816 0.2042469 -0.11271635 0.9493299
9 -0.43301073 -0.1442145 0.95462153 3.3123012
10 0.95831249 0.3290120 -1.71252687 3.3632555
NO_Of_UNITS_TV FAMILY_FRIENDLY_TV FREEHOLD_TV Local_R2
1 0.480589953 -0.06902748 2.621347 0.8846744
2 -1.380026395 2.69655779 3.045280 0.8899773
3 -0.045279967 0.04058290 1.197050 0.8947007
4 -0.446007570 14.31276425 8.771149 0.9073605
5 0.141120178 0.36714544 1.557983 0.9510057
6 -0.355762335 0.60891234 1.260522 0.9247586
7 -0.001731033 2.78285441 2.243875 0.8310458
8 0.089093858 0.32346758 1.602012 0.9463936
9 1.534793921 -0.13828365 2.285410 0.8380365
10 -0.082251138 -0.53369623 1.033819 0.9080753
geometry
1 POINT (22085.12 29951.54)
2 POINT (25656.84 34546.2)
3 POINT (23963.99 32890.8)
4 POINT (27044.28 32319.77)
5 POINT (41042.56 33743.64)
6 POINT (39717.04 32943.1)
7 POINT (28419.1 33513.37)
8 POINT (40763.57 33879.61)
9 POINT (23595.63 28884.78)
10 POINT (24586.56 33194.31)Show the code
Rows: 1,436
Columns: 77
$ POSTCODE <dbl> 118635, 288420, 267833, 258380, 467169, 466472…
$ SELLING_PRICE <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ AREA_SQM <dbl> 309, 290, 248, 127, 145, 139, 218, 141, 165, 1…
$ AGE <dbl> 30, 32, 33, 7, 28, 22, 24, 24, 27, 31, 17, 22,…
$ PROX_CBD <dbl> 7.941259, 6.609797, 6.898000, 4.038861, 11.783…
$ PROX_CHILDCARE <dbl> 0.16597932, 0.28027246, 0.42922669, 0.39473543…
$ PROX_ELDERLYCARE <dbl> 2.5198118, 1.9333338, 0.5021395, 1.9910316, 1.…
$ PROX_URA_GROWTH_AREA <dbl> 6.618741, 7.505109, 6.463887, 4.906512, 6.4106…
$ PROX_HAWKER_MARKET <dbl> 1.76542207, 0.54507614, 0.37789301, 1.68259969…
$ PROX_KINDERGARTEN <dbl> 0.05835552, 0.61592412, 0.14120309, 0.38200076…
$ PROX_MRT <dbl> 0.5607188, 0.6584461, 0.3053433, 0.6910183, 0.…
$ PROX_PARK <dbl> 1.1710446, 0.1992269, 0.2779886, 0.9832843, 0.…
$ PROX_PRIMARY_SCH <dbl> 1.6340256, 0.9747834, 1.4715016, 1.4546324, 0.…
$ PROX_TOP_PRIMARY_SCH <dbl> 3.3273195, 0.9747834, 1.4715016, 2.3006394, 0.…
$ PROX_SHOPPING_MALL <dbl> 2.2102717, 2.9374279, 1.2256850, 0.3525671, 1.…
$ PROX_SUPERMARKET <dbl> 0.9103958, 0.5900617, 0.4135583, 0.4162219, 0.…
$ PROX_BUS_STOP <dbl> 0.10336166, 0.28673408, 0.28504777, 0.29872340…
$ NO_Of_UNITS <dbl> 18, 20, 27, 30, 30, 31, 32, 32, 32, 32, 34, 34…
$ FAMILY_FRIENDLY <dbl> 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0…
$ FREEHOLD <dbl> 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1…
$ LEASEHOLD_99YR <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ LOG_SELLING_PRICE <dbl> 14.91412, 15.17135, 15.01698, 15.26243, 14.151…
$ MLR_RES <dbl> -1489099.55, 415494.57, 194129.69, 1088992.71,…
$ Intercept <dbl> 2050011.67, 1633128.24, 3433608.17, 234358.91,…
$ AREA_SQM.1 <dbl> 9561.892, 16576.853, 13091.861, 20730.601, 672…
$ AGE.1 <dbl> -9514.634, -58185.479, -26707.386, -93308.988,…
$ PROX_CBD.1 <dbl> -120681.94, -149434.22, -259397.77, 2426853.66…
$ PROX_CHILDCARE.1 <dbl> 319266.925, 441102.177, -120116.816, 480825.28…
$ PROX_ELDERLYCARE.1 <dbl> -393417.795, 325188.741, 535855.806, 314783.72…
$ PROX_URA_GROWTH_AREA.1 <dbl> -159980.203, -142290.389, -253621.206, -267929…
$ PROX_MRT.1 <dbl> -299742.96, -2510522.23, -936853.28, -2039479.…
$ PROX_PARK.1 <dbl> -172104.47, 523379.72, 209099.85, -759153.26, …
$ PROX_PRIMARY_SCH.1 <dbl> 242668.03, 1106830.66, 571462.33, 3127477.21, …
$ PROX_SHOPPING_MALL.1 <dbl> 300881.390, -87693.378, -126732.712, -29593.34…
$ PROX_BUS_STOP.1 <dbl> 1210615.44, 1843587.22, 1411924.90, 7225577.51…
$ NO_Of_UNITS.1 <dbl> 104.8290640, -288.3441183, -9.5532945, -161.35…
$ FAMILY_FRIENDLY.1 <dbl> -9075.370, 310074.664, 5949.746, 1556178.531, …
$ FREEHOLD.1 <dbl> 303955.61, 396221.27, 168821.75, 1212515.58, 3…
$ y <dbl> 3000000, 3880000, 3325000, 4250000, 1400000, 1…
$ yhat <dbl> 2886531.8, 3466801.5, 3616527.2, 5435481.6, 13…
$ residual <dbl> 113468.16, 413198.52, -291527.20, -1185481.63,…
$ CV_Score <dbl> 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0…
$ Stud_residual <dbl> 0.38207013, 1.01433140, -0.83780678, -2.846146…
$ Intercept_SE <dbl> 516105.5, 488083.5, 963711.4, 444185.5, 211962…
$ AREA_SQM_SE <dbl> 823.2860, 825.2380, 988.2240, 617.4007, 1376.2…
$ AGE_SE <dbl> 5889.782, 6226.916, 6510.236, 6010.511, 8180.3…
$ PROX_CBD_SE <dbl> 37411.22, 23615.06, 56103.77, 469337.41, 41064…
$ PROX_CHILDCARE_SE <dbl> 319111.1, 299705.3, 349128.5, 304965.2, 698720…
$ PROX_ELDERLYCARE_SE <dbl> 120633.34, 84546.69, 129687.07, 127150.69, 327…
$ PROX_URA_GROWTH_AREA_SE <dbl> 56207.39, 76956.50, 95774.60, 470762.12, 47433…
$ PROX_MRT_SE <dbl> 185181.3, 281133.9, 275483.7, 279877.1, 363830…
$ PROX_PARK_SE <dbl> 205499.6, 229358.7, 314124.3, 227249.4, 364580…
$ PROX_PRIMARY_SCH_SE <dbl> 152400.7, 165150.7, 196662.6, 240878.9, 249087…
$ PROX_SHOPPING_MALL_SE <dbl> 109268.8, 98906.8, 119913.3, 177104.1, 301032.…
$ PROX_BUS_STOP_SE <dbl> 600668.6, 410222.1, 464156.7, 562810.8, 740922…
$ NO_Of_UNITS_SE <dbl> 218.1258, 208.9410, 210.9828, 361.7767, 299.50…
$ FAMILY_FRIENDLY_SE <dbl> 131474.73, 114989.07, 146607.22, 108726.62, 16…
$ FREEHOLD_SE <dbl> 115954.0, 130110.0, 141031.5, 138239.1, 210641…
$ Intercept_TV <dbl> 3.9720784, 3.3460017, 3.5629010, 0.5276150, 1.…
$ AREA_SQM_TV <dbl> 11.614302, 20.087361, 13.247868, 33.577223, 4.…
$ AGE_TV <dbl> -1.6154474, -9.3441881, -4.1023685, -15.524301…
$ PROX_CBD_TV <dbl> -3.22582173, -6.32792021, -4.62353528, 5.17080…
$ PROX_CHILDCARE_TV <dbl> 1.000488185, 1.471786337, -0.344047555, 1.5766…
$ PROX_ELDERLYCARE_TV <dbl> -3.26126929, 3.84626245, 4.13191383, 2.4756745…
$ PROX_URA_GROWTH_AREA_TV <dbl> -2.846248368, -1.848971738, -2.648105057, -5.6…
$ PROX_MRT_TV <dbl> -1.61864578, -8.92998600, -3.40075727, -7.2870…
$ PROX_PARK_TV <dbl> -0.83749312, 2.28192684, 0.66565951, -3.340617…
$ PROX_PRIMARY_SCH_TV <dbl> 1.59230221, 6.70194543, 2.90580089, 12.9836104…
$ PROX_SHOPPING_MALL_TV <dbl> 2.753588422, -0.886626400, -1.056869486, -0.16…
$ PROX_BUS_STOP_TV <dbl> 2.0154464, 4.4941192, 3.0419145, 12.8383775, 0…
$ NO_Of_UNITS_TV <dbl> 0.480589953, -1.380026395, -0.045279967, -0.44…
$ FAMILY_FRIENDLY_TV <dbl> -0.06902748, 2.69655779, 0.04058290, 14.312764…
$ FREEHOLD_TV <dbl> 2.6213469, 3.0452799, 1.1970499, 8.7711485, 1.…
$ Local_R2 <dbl> 0.8846744, 0.8899773, 0.8947007, 0.9073605, 0.…
$ coords.x1 <dbl> 22085.12, 25656.84, 23963.99, 27044.28, 41042.…
$ coords.x2 <dbl> 29951.54, 34546.20, 32890.80, 32319.77, 33743.…
$ geometry <POINT [m]> POINT (22085.12 29951.54), POINT (25656.… Min. 1st Qu. Median Mean 3rd Qu. Max.
171347 1102001 1385528 1751842 1982307 13887901 Visualising local R2
Local R2 range between 0.0 and 1.0 and indicate how well the local regression model fits observed y values. Very low values indicate the local model is performing poorly. Mapping the Local R2 values to see where GWR predicts well and where it predicts poorly may provide clues about important variables that may be missing from the regression model.
tmap mode set to interactive viewingShow the code
Warning: The shape mpsz is invalid (after reprojection). See sf::st_is_validtmap mode set to plottingBy coefficient estimates
tmap mode set to interactive viewingShow the code
tmap_options(check.and.fix = TRUE)
AREA_SQM_SE <- tm_shape(mpsz)+
tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +
tm_dots(col = "AREA_SQM_SE",
border.col = "gray60",
border.lwd = 1) +
tm_view(set.zoom.limits = c(11,14))
AREA_SQM_TV <- tm_shape(mpsz)+
tm_polygons(alpha = 0.1) +
tm_shape(condo_resale.sf.adaptive) +
tm_dots(col = "AREA_SQM_TV",
border.col = "gray60",
border.lwd = 1) +
tm_view(set.zoom.limits = c(11,14))
tmap_arrange(AREA_SQM_SE, AREA_SQM_TV,
asp=1, ncol=2,
sync = TRUE)Warning: The shape mpsz is invalid (after reprojection). See sf::st_is_valid
Warning: The shape mpsz is invalid (after reprojection). See sf::st_is_validtmap mode set to plottingBy planning region
